Concept explainers
Use a calculator or computer to make a table of values of left and right Riemann sums Ln and Rn for the
To Calculate: The left Riemann sum
To Find: The two numbers which the integral
To Explain: the statement for the integral
Answer to Problem 16E
The value of right Riemann sums
The value of left Riemann sums
The value of the integral lies between
No, similar statement cannot be made for the integral
Explanation of Solution
Given:
The integral function as
Number of rectangles
Calculation:
Show the Equation of the integral as follows:
Consider the value of the function
Calculate the value of the function
Substitute 0 for x in Equation (6).
Substitute 2 for x in Equation (6).
The function
For
Find the width
Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.
The limits of the integral
Substitute
Calculate the right Riemann Sum for
The right endpoints are
The expression to find the right Riemann sum
Here, the right endpoint height of first rectangle is
Calculate the value of
Substitute
Substitute
Substitute
Substitute
Substitute
Calculate the right Riemann sum using calculator.
Substitute
The value of right Riemann Sum for
Similarly calculate the right Riemann sum for
Tabulate the values of right Riemann sum for
n | |
5 | 0.68479 |
10 | 0.68115 |
50 | 0.19652 |
100 | 0.09956 |
Table 1
Calculate the left Riemann Sum for
The left endpoints are
The expression to find the left Riemann sum
Here, the left endpoint height of first rectangle is
Calculate the value of
Substitute
Substitute
Substitute
Substitute
Substitute
Calculate the left Riemann sum using calculator.
Substitute
The value of left Riemann Sum for
Similarly calculate the left Riemann sum for
Tabulate the values of left Riemann sum for
n | |
5 | 1.07746 |
10 | 0.80758 |
50 | 0.19809 |
100 | 0.09976 |
Table 2
Refer to Table 1 and Table 2.
The value of the integral lies as shown below:
The value of the integral lies between
whether similar statement can be made for the integral
Consider the value of the function
Calculate the value of the function
Substitute
Substitute 0 for x in Equation (6).
Substitute 2 for x in Equation (6).
No, similar statement cannot be made for the integral
Chapter 5 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning